Brownian motion in twisted domains

The tail behavior of a Brownian motion's exit time from an unbounded domain depends upon the growth of the ``inner radius'' of the domain. In this article we quantify this idea by introducing the notion of a twisted domain in the plane. Roughly speaking, such a domain is generated by a planar curve as follows. As a traveler proceeds out along the curve, the boundary curves of the domain are obtained by moving out $\pm g(r)$ units along the unit normal to the curve when the traveler is $r$ units away from the origin. The function $g$ is called the growth radius. Such domains can be highly nonconvex and asymmetric. We give a detailed account of the case $g(r) = \gamma r^p$, $0<p\le 1$. When $p=1$, a twisted domain can reasonably be interpreted as a ``twisted cone.''